A106458 -id:A106458 - cp's OEIS frontend

A159688 Triangle read by rows, denominators of Jakob Bernoulli's "Sums of Powers" triangle.

1, 2, 2, 3, 2, 6, 4, 2, 4, 5, 2, 3, -30, 6, 2, 12, -12, 7, 2, 2, -6, 42, 8, 2, 12, -24, 12, 9, 2, 3, -15, 9, -30, 10, 2, 4, -10, 2, -20, 11, 2, 6, -1, 1, -2, 66, 12, 2, 12, -8, 6, -8, 12, 13, 2, 1, -6, 7, -10, 3, -2730, 14, 2, 12, -60, 28, -20, 12, -420, 15, 2, 6, -30, 18, -10, 6, -90, 6, 16, 2, 4, -24, 12, -16, 12, -24, 4

Offset: 0

Gary W. Adamson, Apr 19 2009

Let the triangle = T. Row sums = 1. Row sums of n-th binomial transform of T = powers of (n-1). Then multiply the results by the partial sum operator, (1; 1,1; 1,1,1; ...) to obtain Bernoulli's "Sums of Powers".

Inserting zeros to account for (n+1) terms per row, right border = Bernoulli numbers: (A106458): (1, 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ...).

			Let row 0 = 1; followed by the corrected table, giving denominators:
   1;
   2, 2;
   3, 2,  6;
   4, 2,  4;
   5, 2,  3, -30;
   6, 2, 12, -12;
   7, 2,  2,  -6, 42;
   8, 2, 12, -24, 12;
   9, 2,  3, -15,  9, -30;
  10, 2,  4, -10,  2, -20;
  11, 2,  6,  -1,  1,  -2, 66;
  ...
The complete triangle with row 0 = 1, along with numerators:
  1;
  1/2,  1/2;
  1/3,  1/2, 1/6;
  1/4,  1/2, 1/4;
  1/5,  1/2, 1/3,  -1/30;
  1/6,  1/2, 5/12, -1/12;
  1/7,  1/2, 1/2,  -1/6,  1/42;
  1/8,  1/2, 7/12, -7/14, 1/12;
  1/9,  1/2, 2/3,  -7/15, 1/2, -3/20;
  1/10, 1/2, 3/4,  -7/10, 1/2, -3/20;
  1/11, 1/2, 5/6,  -1/1,  1/1, -1/2,  5/66;
  ...

Jakob Bernoulli, "Ars conjectandi", posthumously published in 1713, in which Bernoulli gives the table "Summae Potestatum (Sums of Powers) [cf. Young, p. 86].
Robert M. Young, "Excursions in Calculus", MAA, 1992.

Seiichi Manyama, Rows n = 0..200, flattened
Wikipedia, Faulhaber's formula.

Cf. A106458.

f[n_, x_] := f[n, x] = ((x+1)^(n+1) - 1)/(n+1) - Sum[Binomial[n+1, k]*f[k, x], {k, 0, n-1}]/(n+1); f[0, x_] := x; row[n_] := CoefficientList[f[n, x], x] // Reverse // (Sign[#]*Denominator[#])& // DeleteCases[#,0]&; Table[row[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Dec 29 2012 *)

Extended to 15 rows by Jean-François Alcover, Dec 29 2012

A140814 a(0)=3, a(n)=A002445(n) for n >= 1.

Original entry on oeis.org

3, 6, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, 1806, 690, 282, 46410, 66, 1590, 798, 870, 354, 56786730, 6, 510, 64722, 30, 4686, 140100870, 6, 30, 3318, 230010, 498, 3404310, 6, 61410, 272118, 1410, 6

Offset: 0

Paul Curtz, Jul 16 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A027762, A140816.

[3] cat [Denominator(Bernoulli(2*n)): n in [1..60]]; // Vincenzo Librandi, Nov 04 2018

Join[{3}, Table[Denominator[BernoulliB[2 n]], {n, 60}]] (* Vincenzo Librandi, Nov 04 2018 *)

a(n) = A106458(2*n) + A106458(2*n+1).

a(n) = A027762(n) for n >= 1. - Georg Fischer, Nov 03 2018

Edited and extended by R. J. Mathar, Jul 29 2008

cp's OEIS Frontend

A159688 Triangle read by rows, denominators of Jakob Bernoulli's "Sums of Powers" triangle.

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